C « 9 3 
X. On the Integration of certain differential Expressions , with 
which Problems in physical Astronomy are connected, &c. By 
Robert Woodhouse, A. M. F. R. S. Fellow of Cains College . 
Read April 12, 1804. 
In analytical investigation, two important objects present them- 
selves : the concise and unambiguous expression of the condi- 
tions of a problem in algebraic language ; and the reduction of 
such expression into forms commodious for arithmetical com- 
putation. 
If the introduction of the new calculi, as they have been 
called, has extended the bounds of science, it has enormously 
increased its difficulties, in their number and magnitude. The 
differential forms that can be completely integrated, occur in few 
problems only, and those of small moment. In physical astro- 
nomy, the investigations give rise to differential expressions, 
which call forth all the resources of the analytic art, even for 
their approximate integration. 
For the integration of differential expressions that, by the 
process of taking the differential, can be derived from no finite 
algebraic form, recourse is had to infinite series : thus, if the 
expression be dx .fx, and there is no quantity Fx, such that 
dx .fx — d (Fx) : fx is put =/((x — a ) -f a) —f (x — a) 
+ d/ (x — a) . a + D 2 / ( x — a) a* -f &c. and f dx .fx — 
fdx .f(x ~ a) 4* fdx .Df(x-r-a) + Jdx . — a ) &c. or, 
